Monthly Archives: July 2012

Last week I spent my spring break loitering at the math teachers conference in Anaheim. I learned so much it’s been hard to contain it all. Here are some of my observations:.

I learned that no-one checks your badge to see if it’s counterfeit.

If you stay at the hotels, the daily parking is $16 where if you only drive in for the day it is $8. And if you park in Garden Grove and walk, it’s free. I would think the nightly hotel bill of $160 would offset some of the parking expenses.

I was pleasantly surprised when the parking attendant remembered me from my visit two days earlier.

If you buy a hot dog and soda at the Anaheim Convention center it costs $9.50. I would think the $180 registration fee would offset some of the hot dog expenses.

I learned that $180 is actually inexpensive for a conference of this size and that the many vendors probably made up the difference.

You can get a lot of free stuff from vendors who want you to spend your school’s money on a lot of expensive stuff.

There are schools that can’t afford the expensive stuff so their teachers spend their own money on less effective substitutes.

I learned that far too many teachers spend their own money on less effective substitutes.

A cheerleading competition was scheduled next door. (I assume it was to cheer on the math teachers.)

People came as far as Australia. (To the math convention. I don’t know about the cheerleading one.)

I met a principal that wants me to move to New Mexico. (I assume it was to teach there.)

I met a teacher from Compton, California that doesn’t think merit pay based on student performance is fair when she has Algebra classes with 40 plus at-risk students while other districts have more motivated classes of thirty students or less.

She told me that she is considering quitting.

I met a surprisingly large number of people who were looking for the boat show. (They must have walked in from Garden Grove and didn’t see the directions.)

I discovered there is free wireless available on the second floor of the Anaheim Convention Center but not anywhere else.

I did learn some math teacher stuff but won’t bore you with the details…

Liberal arts education is to make people into good citizens, not into good workers. They are to acquaint you with the intellectual achievements of humankind. That is why we read the Iliad, why we watch a performance of Hamlet, why we learn about the history of ancient Greece, and, yes, why we study algebra.

We live in a technological society. Not learning algebra in the public school system means those kids will not be prepared, will not be qualified, to do anything in science and engineering. I’m serious: if you don’t know algebra, you can’t do basic quantitative chemistry, and if you can’t do that, you can’t do biology. At all. Not the molecular/biochemical/bench side, not the ecological/evolutionary/field side. You can’t do physics, that’s for sure. Forget math and statistics. If you’re not capable of grasping statistics, forget psychology, too.

I’ll go farther and hazard a guess that poli sci, like the other social sciences, is much more empirical and quantitative now than it was then and that Dr. Hacker would be hard put to earn a doctorate in his own field nowadays.

Economists have shown that cognitive skills–especially math and science–are robust predictors of individual income, of a country’s economic growth, and of the distribution of income within a country (e.g. Hanushek & Kimko, 2000; Hanushek & Woessmann, 2008).

A patient who understands graphs and inequalities will have a much better control of their health than someone who has trouble with even basic arithmetic. This is not a case of mathematics being a prerequisite for a job, the usual argument for numeracy. This is mathematics making a difference in your own health.

On that note, here’s commenter Ben: Is English literature necessary? I never ran into Moby Dick during my career. Is Gym class necessary? I never had to do a pull up. Is Physics necessary? When was I called upon to do vector analysis? Is Biology necessary? I think only French Chefs have to dissect frogs in their careers. It must be tough to teach. What’s the point. Walmart has all the stuff we’ll ever need. Supersize that happy meal. We’re just turning into a bunch of consumers anyway, lets just stop pretending that we need to know anything.

Few pre-college math teachers majored or even minored in math, and until more teachers do, improvements will be hard to come by. Ironically, it seems that people who have mastered “useless” algebra and other higher math topics tend to get jobs that pay more than middle school math teachers earn. I have the utmost respect for people with math degrees who choose to teach in spite of the poor pay and discipline problems, but few people make that choice. Math education needs help, but Hacker’s suggestions throw out the baby with the bathwater.

The advent of the Internet and the emergence of quantifiable data is making math skills critical to many white-collar professions; marketers and public relations staffers, for example, have to understand the arcane aspects of statistics in order to analyze data on ad campaigns, while reporters and editorialists need stronger math skills as well.

To say that we are failing and need to do something about it is absolutely correct. To say we need to remove it from our path as a hindrance rather than overcoming it with improved teaching methods, enhanced mathematics programs, and funding in general is a mistake.

When students say they have difficultly with algebra, that’s usually not the entire story. Typically, that means they have also trouble with arithmetic. There’s a reason why the ability to do long division is correlated with long-term mathematics performance: you have to master the basics.

My girlfriend is a fine artist, with an MFA in sculpting from a school . . . in New York City. Earlier this year, she was the recipient of a month-long artist residency in Taiwan where she put together an outdoor installation in knitted recycled plastic as part of an exhibit on environmental themes. She has a fairly high proficiency at math . . . and this gets used routinely in her career. She has to estimate volumes of complicated shapes she’s planning to put together, so as to procure materials . . . She has to do calculations with money so as to set budgets . . . She has to estimate time for projects that might last many months. At some point she generated a calculation for people, time, and material to cover the Eiffel Tower in tiny crocheted plastic leaves (a long-term goal).

Some people have questioned whether our students should be required to learn algebra. Andrew Hacker at the New York Times points out how many students struggle and fail algebra. The commenters, fortunately, point out all of his flawed logic. (Someone should summarize all of the great comments explaining why algebra should be required.)

Richard Cohen at the Washington Post makes the idiotic argument: “You will never need to know algebra. I have never once used it and never once even rued that I could not use it.”

If I had never heard a poem or listened to a symphony or read a novel or visited Independence Hall, I could probably dumbly write that I don’t miss literature, music, or history…never heard of ‘em. Don’t need ‘em. Bugger all you eggheads pushing your useless ‘knowledge’ on me!

This reminds me of when I was student teaching. The supervisor of my student teaching supervisor, a supposedly educated man with a PhD, said: I never took Calculus and have never regretted it. There has never been a time when I wish I had learned calculus.

What an idiot! (Sorry for being rude but it’s true.) If you don’t know a subject, how do you know if you couldn’t use it?

Everyone is impacted by economics. As a discipline, economics is based in calculus. I remember being in a general-ed economics class and sitting through long convoluted verbal explanations of marginal something-or-other. I thought to myself, “It’s the derivative stupid!” They are simply describing the change in the something-or-other given a small change in the input. (Sorry I don’t remember the details.) If everyone in the room had learned calculus, we could have covered a 2 hour lecture in 30 minutes.

So when could you use calculus? Everywhere! If, and only if, you understand it. The same is true for algebra.

Update: I just read a post by Chad Orzel, also at scienceblogs, that calls bullshit on the acceptance of innumeracy by intellectuals.

Update II: I wrote this thinking all of the articles were recent. (They showed up today in my news reader and I know this is currently being discussed elsewhere.) It turns out they are up to 6 years old. So please forgive my use of present tense. All of the arguments are still valid. Note to self: Check the publication dates on articles sent by Zite…

I’ve been reading up on Dan Meyers “Ladder of Abstraction.” It turns out he didn’t invent it. I think the phrase was first used by H. I. Hayakawa. And there are obviously different versions of it. The most concrete examples deal with language. (Yes it’s ironic I used the word concrete. One could argue that Hayakawa’s entire ladder is on the lowest rungs of the math ladders.)

Abstracting from Cow to Wealth:

English teachers have it so easy… Except for the reading essays part.

Another version was described by Michael Matassa and Frederick Peck at the ICME-12 conference. It refers to the Iceberg Metaphor and shows students going three stages: Concrete, Preformal, Formal. (I’m not sure if they call it The Ladder of Abstraction.)

I’ve been trying to think of some universal system for understanding and/or naming the different rungs.

You may ask: doesn’t Concrete, Preformal, Formal count? It’s a start. The problem is that, beyond algebra, most math lives in the formal category. For example, you may have a student that totally understands the equations for projectile motion. But when you try to generalize to all parabolas the student becomes completely lost. Both of these live in the “formal” section of the ladder but the second is clearly on a higher rung.

I’ve been trying to think of a way to break it down into more explicit rungs on the ladder. But as pointed out by Dan, it depends on the question you’re asking.

He starts with the same picture and shows how it can lead to different ladders with different rungs. So instead of trying to create a universal ladder it might be more useful to see how the idea of the ladder works with different problems.

So I’m in the process of creating different ladders which all start with a cow as the bottom rung.

While you wait in anticipation you can check out this version which looks at design…

It really is a cool system. You can have students do work on their calculator and wirelessly send you the results. You can give quizzes, get instant feedback and adjust your instruction. You can also display an image and have students interact with it, which we did in the workshop.

To demonstrate linear regression the TI representative displayed some image similar to:

The participants were asked to plot a point anywhere on the bridge’s fence line. These points, as well as their calculator number, showed up on the image. Most of the teacher-participants were very obedient and picked a point exactly on the line. Somebody wanted to see what would happen if he picked a point which wasn’t even close to the fence line.

The resulting picture looked like…

The TI guy happily showed us how to find and graph the equation which looked like:

Oops…

After a long pause the presenter said, “That’s never happened before…”

I was starting to feel bad for the presenter when some annoyingly helpful teacher pointed out the point in the corner. Another teacher noticed the calculator number next to the point…

I was quietly hiding my calculator number when the presenter commented on turning something like this into a teaching moment…

When are we ever going to use this? I’m sure other teachers hear this but it seems to be especially prevalent in math classes.

This question is frustrating for two reasons. 1) There is an implied “if you can’t answer then I don’t have to learn it.” 2) I could come up with an infinite number of answers and they, probably, would not be adequate. Students invariably say yeah yeah, but I don’t planning on doing that.

But it is a fair question. I’ve created this blog, in part, to hone my answers. Here are my current top ten reasons. (They are not in any particular order except, perhaps, #1 and 10.)

Learning math can help you….

10. Prepare for a career.

In my humble opinion, this is the least important reason to learn math….

9. Develop problem solving skills.

You can only develop problem solving skills by …wait for it… solving problems. I don’t mean problems where the teacher shows you an example and you do 20 problems just like it. I am referring to problems you’ve never seen before. Math gives you practice in organizing what you know, rearranging information, testing hypotheses, etc. These are the same skills you can use in everyday life. Unfortunately, most teachers (including myself… There’s that stupid test each year) take the problem solving element out of math class.

8. Increase your capacity to think analytically

Without props, without manipulatives…. Just with your brain…

7. Be able to learn science

6. Argue better

5. Be less gullible

Astrology is completely bogus. This can be demonstrated. If you don’t understand or believe the reasons (or can’t sit through an explanation), you’re missing some of the skills mentioned above and your math teachers should be fired, (except for the fact that learning is, ultimately, the students’ responsibility.)

4. Distinguish us from animals

What makes humans different? Build tools? Nope, certain apes do it, even ants. Develop language? Nope, dolphins, birds, and primates beat us to it. Use money? Nope. One reason is our ability to think and communicate abstract ideas. One language for doing so is mathematics. (To be fair there are many others: art, music, literature, etc.)

3. Better appreciate the history of our civilization

Wait, what?

The industrial revolution was made possible because of thermodynamics which require an understanding of differential equations which is the main sequel to Newton’s calculus. The exploration of the world happened because the Egyptians knew the earth was round. etc.

2. So you don’t look like an idiot

And to show off at parties…

1. If you understand math…. And can read, you can teach yourself anything….

So when I ask you, “why is math important.” if you say, “to count my money.” I will start banging my head against the wall.

Math teachers throughout time have said, “math is important to help you develop problem solving skills.” And then they show you how to solve the problem and you know how to solve THAT problem. When faced with a new problem you’re just as lost.

But that’s not the problem.

So we develop quasi real world problems that look open-ended… But, in reality, we expect a certain outcome from the students. We expect them to choose a certain strategy and come up with a particular answer, or range of possible answers. If they don’t, we help them by giving them a strategy.

That’s still not the problem.

The problem is the standards. We look at a content standard and say let’s create a problem that utilizes that standard. We give it to the student and they solve it a different way. And we adjust and give the next students steps to solving the problem. Now we’re back to helping the students more than we should because we’re overly attached to the standards.

I may be exaggerating but… “real world” problem solving is not compatible with content based standards such as California’s.

Developing problem solving skills should be the goal, not understanding the difference between equations of ellipses and hyperbolas.